Answer:
(d²x/dt²) + 2*(dx/dt) + 5*x = 16 cos 2t + 4 sin 2t
Step-by-step explanation:
Given
f(t) = 16 cos 2t + 4 sin 2t
Let m be the mass attached, let k be the spring constant and let β be the positive damping constant. The Newton's Second Law for the system is
m*d²x/dt² = - k*x - β*dx/dt + f(t)
where x(t) is the displacement from the equilibrium position and f(t) is the external force. The equation can be transformed into
(d²x/dt²) + (β/m)*(dx/dt) + (k/m)*x = (1/m)*f(t) (i)
a) Let's determine the equation of motion. Put m = 1 slug, k = 5 lb/ft, β = 2 and f(t) = 16 cos 2t + 4 sin 2t into equation (i) to get the differential equation
for x(t):
(d²x/dt²) + (2/1)*(dx/dt) + (5/1)*x = (1/1)*(16 cos 2t + 4 sin 2t)
⇒ (d²x/dt²) + 2*(dx/dt) + 5*x = 16 cos 2t + 4 sin 2t
